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Aseev launched a new branch of functional analysis by introducing the theory of quasilinear spaces in the framework of the topics of norm, bounded quasilinear operators and functionals (Aseev (1986)). Furthermore, some quasilinear counterparts of classical nonlinear analysis that lead to such result as Frechet derivative and its applications were examined deal with. This pioneering work causes a lot of results in such applications such as (Rojas-Medar et al. (2005), Talo and Başar (2010), and Nikol'skiĭ (1993)). His work has motivated us to introduce the concept of quasilinear inner product spaces. Thanks to this new notion, we obtain some new theorems and definitions which are quasilinear counterparts of fundamental definitions and theorems in linear functional analysis. We claim that some new results related to this concept provide an important contribution to the improvement of quasilinear functional analysis.

The theory of quasilinear space was introduced by Aseev [

As known, the theory of inner product space and Hilbert spaces play a fundamental role in functional analysis and its applications such as integral and differential equations, approximation theory, linear and nonlinear stability problems, and bifurcation theory.

We know that any inner product space is a normed space and any normed space is a particular class of normed quasilinear space. Hence, this relation and Aseev's work motivated us to examine quasilinear counterparts of inner product space in classical analysis. Thus, we introduce the concept of quasilinear inner product space as a new structure. Moreover, we obtain some definitions and results related to this notion which provide us with improving the elements of the quasilinear functional analysis.

The definition of quasi-inner product function is extended by classical definition of inner product function. It is normal to expect that inner product which is defined by quasilinear space is supposed to be given by means of a partial order relation, just as in the method defining quasilinear normed spaces. Then we clearly observe from these definitions that the concept of quasilinear inner product space is a generalization of inner product space. While working on this new concept, we noticed that there were some differences related to analysis as different from classical case. For example, the convergence inproof ofcontinuity of quasi-inner product function according to the Hausdorff metric of the quasilinear space leads to slight differences in details of the proof.

In another important part of this work, we introduce the notion of Hilbert quasilinear space, inner product

In this paper we aim to give a contribution to the studies on quasilinear spaces by introducing the notion of quasilinear inner product spaces.

Let us start this section by introducing the definition of quasilinear spaces and some of their basic properties given by Aseev [

Aseev proceeds in a similar way to linear functional analysis on quasilinear spaces by introducing the notions of the norm and quasilinear operators and functionals. Further, he presented some results which are quasilinear counterparts of fundamental definitions and theorems in linear functional analysis.

A set

There exists an element

A linear space is a QLS with the partial order relation “=.” Perhaps the most popular example which is not a linear space is the set of all closed intervals of real numbers with the inclusion relation “

An element

Suppose that any element

In a real linear space, equality is the only way to define a partial order such that conditions (1)–(13) hold.

It will be assumed in what follows that

Suppose that

There exists an element

We note that if

Let

Let

Let

if

if for any

A quasilinear space

Let

Since

The operations of algebraic sum and multiplication by real numbers are continuous with respect to the Hausdorff metric. The norm is continuous function with respect to the Hausdorff metric.

(i) Suppose that

(ii) Suppose that

(iii) Suppose that

Let

Let

A normed quasilinear space

(a) If

(b) Suppose that

The

Let

A quasilinear space with a quasi-inner product is called a quasilinear inner product space.

Indeed, since it is easy to verify that the conditions (

Let

Suppose that for any

This is a contradiction. Thus

Every quasilinear inner product space

Since it can be seen easily that the conditions (i)–(iii) hold we give only the proof of the conditions (iv) and (v).

It is routine to show that a norm on a quasilinear inner product space satisfies the parallelogram equality

We conclude that if a norm does not satisfy the parallelogram equality, it cannot be obtained from a quasilinear inner product by the use of (

Now let us define an inner product function on the quasilinear space

It can be easily seen that Schwartz inequality

The proof of the following result is similar to its classical linear counterpart. But we note that the convergence of a sequence in a quasilinear space is different from the convergence of a sequence in a linear space. Here, convergence is according to the Hausdorff metric of quasilinear space. Because of this relation there are slight differences in details of proof here.

If in a quasilinear inner product space

Since

Let

Let

From (

Let

for every

for every

The proof of part (i) is obvious. If

A quasilinear inner product space

We recall that any normed linear space cannot be an

Let

Now, we show that

If

A quasilinear inner product space is called Hilbert quasilinear space, if it is complete according to the Hausdorff metric.

A quasilinear inner product space

Let

An element

An

(a)

(b) Let

For orthogonal elements

Let

For any subset

If

This theorem implies that

Suppose that

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors thank the referee for his/her valuable comments and helpful suggestions.